Twist-equivalence conjecture for characters of complex reflection groups

Let $G(m,p,n)$ be a complex reflection group, let $J_1$ and $J_{-\mathbf i}$ be the character-twisting maps considered in the paper, and let $\chi$ be a character of $G(m,p,n)$. Twist-equivalence conjecture. If $\frac{m}{p}$ is even, then

$$\chi\circ J_1=\chi\circ J_{-\mathbf i}.$$

This conjecture extends the observed equality of the twists for the irreducible characters of $B_n=G(2,1,n)$ to all groups $G(m,p,n)$ with $\frac{m}{p}$ even.